Question 1. Let a: E→ E be defined by a(x, y) = (3x1,x + 2y + 1).(a) Prove that a is a transformation.(b) Let be the line given by the equation 2x + y − 1 = 0. Find the Cartesian equation of a(l).(c) Is a an involution or isometry?

Answer 1

(a) To prove that a is a transformation, we need to show that it satisfies two properties:

i. Closure under the operation: For any (x, y) in E, a(x, y) must be in E.

ii. Preserves addition and scalar multiplication: For any (x, y) and (u, v) in E, a(x, y) + a(u, v) = a(x + u, y + v), and for any scalar k, a(kx, ky) = ka(x, y).

Let's verify these properties:

i. For any (x, y), a(x, y) = (3x, x + 2y + 1), and both 3x and x + 2y + 1 are real numbers, so a(x, y) is indeed in E.

ii. Addition:

a(x, y) + a(u, v) = (3x, x + 2y + 1) + (3u, u + 2v + 1) = (3x + 3u, x + 2y + 1 + u + 2v + 1) = (3(x + u), (x + u) + 2(y + v) + 2)

This satisfies the addition property.

Scalar Multiplication:

For any scalar k,

a(kx, ky) = (3(kx), kx + 2(ky) + 1) = (k(3x), k(x + 2y + 1))

Since k is a scalar, both k(3x) and k(x + 2y + 1) are real numbers, so this also satisfies the scalar multiplication property.

Therefore, a is indeed a transformation.

(b) The line l is given by 2x + y - 1 = 0. To find the Cartesian equation of a(l), we need to apply the transformation a to the points on the line l.

Let's express y in terms of x from the equation of the line l:

2x + y - 1 = 0

y = -2x + 1

Now, apply the transformation a to (x, y):

a(x, -2x + 1) = (3x, x + 2(-2x + 1) + 1) = (3x, x - 4x + 2 + 1) = (3x, -3x + 3)

So, the Cartesian equation of a(l) is:

3x - 3y + 3 = 0

(c) To determine if a is an involution or isometry, we need more information about the properties of E and whether a preserves distances or has any special properties related to involution.

Answer 2

(a) To prove that a is a transformation, we need to show that it satisfies two properties:

i. Closure under the operation: For any (x, y) in E, a(x, y) must be in E.

ii. Preserves addition and scalar multiplication: For any (x, y) and (u, v) in E, a(x, y) + a(u, v) = a(x + u, y + v), and for any scalar k, a(kx, ky) = ka(x, y).

Let's verify these properties:

i. For any (x, y), a(x, y) = (3x, x + 2y + 1), and both 3x and x + 2y + 1 are real numbers, so a(x, y) is indeed in E.

ii. Addition:

a(x, y) + a(u, v) = (3x, x + 2y + 1) + (3u, u + 2v + 1) = (3x + 3u, x + 2y + 1 + u + 2v + 1) = (3(x + u), (x + u) + 2(y + v) + 2)

This satisfies the addition property.

Scalar Multiplication:

For any scalar k,

a(kx, ky) = (3(kx), kx + 2(ky) + 1) = (k(3x), k(x + 2y + 1))

Since k is a scalar, both k(3x) and k(x + 2y + 1) are real numbers, so this also satisfies the scalar multiplication property.

Therefore, a is indeed a transformation.

(b) The line l is given by 2x + y - 1 = 0. To find the Cartesian equation of a(l), we need to apply the transformation a to the points on the line l.

Let's express y in terms of x from the equation of the line l:

2x + y - 1 = 0

y = -2x + 1

Now, apply the transformation a to (x, y):

a(x, -2x + 1) = (3x, x + 2(-2x + 1) + 1) = (3x, x - 4x + 2 + 1) = (3x, -3x + 3)

So, the Cartesian equation of a(l) is:

3x - 3y + 3 = 0

(c) To determine if a is an involution or isometry, we need more information about the properties of E and whether a preserves distances or has any special properties related to involution.